Theorem zarmxt1 33847

Description: The Zariski topology restricted to maximal ideals is T₁ . (Contributed by Thierry Arnoux, 16-Jun-2024.)

Hypotheses

Ref	Expression
zartop.1	⊢ 𝑆 = (Spec‘𝑅)
zartop.2	⊢ 𝐽 = (TopOpen‘𝑆)
zarmxt1.1	⊢ 𝑀 = (MaxIdeal‘𝑅)
zarmxt1.2	⊢ 𝑇 = (𝐽 ↾t 𝑀)

Assertion

Ref	Expression
zarmxt1	⊢ (𝑅 ∈ CRing → 𝑇 ∈ Fre)

Proof of Theorem zarmxt1

Dummy variables 𝑖 𝑗 𝑘 𝑙 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zartop.1	. . . 4 ⊢ 𝑆 = (Spec‘𝑅)
2		zartop.2	. . . 4 ⊢ 𝐽 = (TopOpen‘𝑆)
3	1, 2	zartop 33843	. . 3 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Top)
4		zarmxt1.1	. . . 4 ⊢ 𝑀 = (MaxIdeal‘𝑅)
5	4	fvexi 6836	. . 3 ⊢ 𝑀 ∈ V
6		zarmxt1.2	. . . 4 ⊢ 𝑇 = (𝐽 ↾t 𝑀)
7		resttop 23045	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ V) → (𝐽 ↾t 𝑀) ∈ Top)
8	6, 7	eqeltrid 2832	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ V) → 𝑇 ∈ Top)
9	3, 5, 8	sylancl 586	. 2 ⊢ (𝑅 ∈ CRing → 𝑇 ∈ Top)
10		eqid 2729	. . . . . . . . . . 11 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
11	10	mxidlprm 33407	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (PrmIdeal‘𝑅))
12	11	ex 412	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (𝑚 ∈ (MaxIdeal‘𝑅) → 𝑚 ∈ (PrmIdeal‘𝑅)))
13	12	ssrdv 3941	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → (MaxIdeal‘𝑅) ⊆ (PrmIdeal‘𝑅))
14	13	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → (MaxIdeal‘𝑅) ⊆ (PrmIdeal‘𝑅))
15		eqid 2729	. . . . . . 7 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
16	14, 4, 15	3sstr4g 3989	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑀 ⊆ (PrmIdeal‘𝑅))
17		sseq2 3962	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑙 → (𝑖 ⊆ 𝑗 ↔ 𝑖 ⊆ 𝑙))
18	17	cbvrabv 3405	. . . . . . . . . . . 12 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑙}
19		sseq1 3961	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝑖 ⊆ 𝑙 ↔ 𝑘 ⊆ 𝑙))
20	19	rabbidv 3402	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑙} = {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑙})
21	18, 20	eqtrid 2776	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑙})
22	21	cbvmptv 5196	. . . . . . . . . 10 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) = (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑙})
23	1, 2, 15, 22	zartopn 33842	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) ∧ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) = (Clsd‘𝐽)))
24	23	adantr 480	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) ∧ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) = (Clsd‘𝐽)))
25	24	simpld 494	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)))
26		toponuni 22799	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) → (PrmIdeal‘𝑅) = ∪ 𝐽)
27	25, 26	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → (PrmIdeal‘𝑅) = ∪ 𝐽)
28	16, 27	sseqtrd 3972	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑀 ⊆ ∪ 𝐽)
29		simpl 482	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑅 ∈ CRing)
30	29	crngringd 20131	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑅 ∈ Ring)
31		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ ∪ 𝑇)
32	6	unieqi 4870	. . . . . . . . . . . 12 ⊢ ∪ 𝑇 = ∪ (𝐽 ↾t 𝑀)
33	31, 32	eleqtrdi 2838	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ ∪ (𝐽 ↾t 𝑀))
34	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝐽 ∈ Top)
35		eqid 2729	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
36	35	restuni 23047	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ⊆ ∪ 𝐽) → 𝑀 = ∪ (𝐽 ↾t 𝑀))
37	34, 28, 36	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑀 = ∪ (𝐽 ↾t 𝑀))
38	33, 37	eleqtrrd 2831	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ 𝑀)
39	38, 4	eleqtrdi 2838	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ (MaxIdeal‘𝑅))
40		eqid 2729	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
41	40	mxidlidl 33400	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (LIdeal‘𝑅))
42	30, 39, 41	syl2anc 584	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ (LIdeal‘𝑅))
43		eqid 2729	. . . . . . . . . 10 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
44	22, 43	zarclssn 33840	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ (LIdeal‘𝑅)) → ({𝑚} = ((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})‘𝑚) ↔ 𝑚 ∈ (MaxIdeal‘𝑅)))
45	44	biimpar 477	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → {𝑚} = ((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})‘𝑚))
46	29, 42, 39, 45	syl21anc 837	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} = ((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})‘𝑚))
47	22	funmpt2 6521	. . . . . . . 8 ⊢ Fun (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
48		fvex 6835	. . . . . . . . . . 11 ⊢ (PrmIdeal‘𝑅) ∈ V
49	48	rabex 5278	. . . . . . . . . 10 ⊢ {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑙} ∈ V
50	49, 22	dmmpti 6626	. . . . . . . . 9 ⊢ dom (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) = (LIdeal‘𝑅)
51	42, 50	eleqtrrdi 2839	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ dom (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
52		fvelrn 7010	. . . . . . . 8 ⊢ ((Fun (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) ∧ 𝑚 ∈ dom (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})) → ((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})‘𝑚) ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
53	47, 51, 52	sylancr 587	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → ((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})‘𝑚) ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
54	46, 53	eqeltrd 2828	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
55	24	simprd 495	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) = (Clsd‘𝐽))
56	54, 55	eleqtrd 2830	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} ∈ (Clsd‘𝐽))
57	38	snssd 4760	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} ⊆ 𝑀)
58	35	restcldi 23058	. . . . 5 ⊢ ((𝑀 ⊆ ∪ 𝐽 ∧ {𝑚} ∈ (Clsd‘𝐽) ∧ {𝑚} ⊆ 𝑀) → {𝑚} ∈ (Clsd‘(𝐽 ↾t 𝑀)))
59	28, 56, 57, 58	syl3anc 1373	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} ∈ (Clsd‘(𝐽 ↾t 𝑀)))
60	6	fveq2i 6825	. . . 4 ⊢ (Clsd‘𝑇) = (Clsd‘(𝐽 ↾t 𝑀))
61	59, 60	eleqtrrdi 2839	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} ∈ (Clsd‘𝑇))
62	61	ralrimiva 3121	. 2 ⊢ (𝑅 ∈ CRing → ∀𝑚 ∈ ∪ 𝑇{𝑚} ∈ (Clsd‘𝑇))
63		eqid 2729	. . 3 ⊢ ∪ 𝑇 = ∪ 𝑇
64	63	ist1 23206	. 2 ⊢ (𝑇 ∈ Fre ↔ (𝑇 ∈ Top ∧ ∀𝑚 ∈ ∪ 𝑇{𝑚} ∈ (Clsd‘𝑇)))
65	9, 62, 64	sylanbrc 583	1 ⊢ (𝑅 ∈ CRing → 𝑇 ∈ Fre)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3394  Vcvv 3436   ⊆ wss 3903  {csn 4577  ∪ cuni 4858   ↦ cmpt 5173  dom cdm 5619  ran crn 5620  Fun wfun 6476  ‘cfv 6482  (class class class)co 7349  Basecbs 17120   ↾_t crest 17324  TopOpenctopn 17325  LSSumclsm 19513  mulGrpcmgp 20025  Ringcrg 20118  CRingccrg 20119  LIdealclidl 21113  Topctop 22778  TopOnctopon 22795  Clsdccld 22901  Frect1 23192  PrmIdealcprmidl 33372  MaxIdealcmxidl 33396  Speccrspec 33829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-ac2 10357  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-rpss 7659  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-oadd 8392  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fi 9301  df-dju 9797  df-card 9835  df-ac 10010  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-rest 17326  df-topn 17327  df-0g 17345  df-topgen 17347  df-mre 17488  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-submnd 18658  df-grp 18815  df-minusg 18816  df-sbg 18817  df-subg 19002  df-cntz 19196  df-lsm 19515  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-cring 20121  df-subrg 20455  df-lmod 20765  df-lss 20835  df-lsp 20875  df-sra 21077  df-rgmod 21078  df-lidl 21115  df-rsp 21116  df-lpidl 21229  df-top 22779  df-topon 22796  df-bases 22831  df-cld 22904  df-t1 23199  df-prmidl 33373  df-mxidl 33397  df-idlsrg 33438  df-rspec 33830

This theorem is referenced by: (None)